Abstract
In the study of concentration properties of isotropic log-concave measures, it is often useful to first ensure that the measure has super-Gaussian marginals. To this end, a standard preprocessing step is to convolve with a Gaussian measure, but this has the disadvantage of destroying small-ball information. We propose an alternative preprocessing step for making the measure seem super-Gaussian, at least up to reasonably high moments, which does not suffer from this caveat: namely, convolving the measure with a random orthogonal image of itself. As an application of this “inner-thickening”, we recover Paouris’ small-ball estimates.
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Acknowledgements
We thank Olivier Guédon and Vitali Milman for discussions. Bo’az Klartag was supported in part by the Israel Science Foundation and by a Marie Curie Reintegration Grant from the Commission of the European Communities. Emanuel Milman was supported by the Israel Science Foundation (grant no. 900/10), the German Israeli Foundation’s Young Scientist Program (grant no. I-2228-2040.6/2009), the Binational Science Foundation (grant no. 2010288), and the Taub Foundation (Landau Fellow).
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Klartag, B., Milman, E. (2012). Inner Regularization of Log-Concave Measures and Small-Ball Estimates. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_15
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DOI: https://doi.org/10.1007/978-3-642-29849-3_15
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